Essay On Snowflakes

1. Introduction
As winter approaches, I get fascinated by the idea of snow. I can still recall the first time I ever saw snow and snowflakes. I especially found snowflakes’ unique shapes and structures absolutely magnificent and beautiful. Therefore, I found myself naturally drawn to explore more about snowflakes, when I had to chose a topic for my mathematical exploration.
The aim of the investigation is to find the patterns of a snowflake and try to make a mathematical sense out of it. In addition, I wanted to explore how to easily calculate the area of a complicated shape as snowflake.
As I began to research how exactly I can approach the question, I encountered fractal geometry, which seemed to explain and address my question. Hence, by using a fractal, Von Koch Island, I have tried to make a “mathematical sense” out of snowflakes by exploring its

Dimensions
Moreover, while Euclidean geometry belong to an integer dimension of either 1, 2, or 3, fractal geometry belongs to so-called fractal dimension. In Euclidean geometry, the number of dimension can be defined as “a set, region, object, or space possesses indicates how many mutually perpendicular directions of movement are possible”; Euclidean geometry obeys the following law:
, where D is dimension, L is linear scaling and S is the resulting increase in size. This is what we are used to.

0-Dimension
1-Dimension
2-Dimension
3-Dimension
Size
Point (0)
Length
(length)2 = Area
(length)3 = Volume

However, fractal geometry does not obey this law. I will talk more about fractal dimensions later in the exploration.
4. Von Koch Island
Von Koch curve is one of the most famous examples of fractal geometry. It was created by a Swedish mathematician, Niels von Koch, in 1906. It looks as following and evidently, the same image is reproduced repeatedly when zoomed in.

…

So how does one form such a shape?

Firstly, have a line. This very beginning can be called the initiator.

0th